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homogeneous harmonic polynomials

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11: Tom H. Koornwinder
Koornwinder has published numerous papers on special functions, harmonic analysis, Lie groups, quantum groups, computer algebra, and their interrelations, including an interpretation of Askey–Wilson polynomials on quantum SU(2), and a five-parameter extension (the Macdonald–Koornwinder polynomials) of Macdonald’s polynomials for root systems BC. … Koornwinder has been active as an officer in the SIAM Activity Group on Special Functions and Orthogonal Polynomials. …
  • 12: Bibliography F
  • P. Falloon (2001) Theory and Computation of Spheroidal Harmonics with General Arguments. Master’s Thesis, The University of Western Australia, Department of Physics.
  • J. L. Fields and Y. L. Luke (1963a) Asymptotic expansions of a class of hypergeometric polynomials with respect to the order. II. J. Math. Anal. Appl. 7 (3), pp. 440–451.
  • J. L. Fields and Y. L. Luke (1963b) Asymptotic expansions of a class of hypergeometric polynomials with respect to the order. J. Math. Anal. Appl. 6 (3), pp. 394–403.
  • J. L. Fields (1965) Asymptotic expansions of a class of hypergeometric polynomials with respect to the order. III. J. Math. Anal. Appl. 12 (3), pp. 593–601.
  • R. Fuchs (1907) Über lineare homogene Differentialgleichungen zweiter Ordnung mit drei im Endlichen gelegenen wesentlich singulären Stellen. Math. Ann. 63 (3), pp. 301–321.
  • 13: Bibliography M
  • I. G. Macdonald (1998) Symmetric Functions and Orthogonal Polynomials. University Lecture Series, Vol. 12, American Mathematical Society, Providence, RI.
  • I. G. Macdonald (2000) Orthogonal polynomials associated with root systems. Sém. Lothar. Combin. 45, pp. Art. B45a, 40 pp. (electronic).
  • T. M. MacRobert (1967) Spherical Harmonics. An Elementary Treatise on Harmonic Functions with Applications. 3rd edition, International Series of Monographs in Pure and Applied Mathematics, Vol. 98, Pergamon Press, Oxford.
  • G. Matviyenko (1993) On the evaluation of Bessel functions. Appl. Comput. Harmon. Anal. 1 (1), pp. 116–135.
  • J. C. P. Miller (1950) On the choice of standard solutions for a homogeneous linear differential equation of the second order. Quart. J. Mech. Appl. Math. 3 (2), pp. 225–235.
  • 14: 18.39 Physical Applications
    §18.39 Physical Applications
    For a harmonic oscillator, the potential energy is given by …
    §18.39(ii) Other Applications
    For physical applications of q -Laguerre polynomials see §17.17. …
    15: 29.18 Mathematical Applications
    §29.18(iii) Spherical and Ellipsoidal Harmonics
    16: 3.7 Ordinary Differential Equations
    If h = 0 the differential equation is homogeneous, otherwise it is inhomogeneous. … … The equations can then be solved by the method of §3.2(ii), if the differential equation is homogeneous, or by Olver’s algorithm (§3.6(v)). … This converts the problem into a tridiagonal matrix problem in which the elements of the matrix are polynomials in λ ; compare §3.2(vi). …
    17: 1.2 Elementary Algebra
    Let α 1 , α 2 , , α n be distinct constants, and f ( x ) be a polynomial of degree less than n . … If m 1 , m 2 , , m n are positive integers and deg f < j = 1 n m j , then there exist polynomials f j ( x ) , deg f j < m j , such that …To find the polynomials f j ( x ) , j = 1 , 2 , , n , multiply both sides by the denominator of the left-hand side and equate coefficients. …
    §1.2(iv) Means
    The geometric mean G and harmonic mean H of n positive numbers a 1 , a 2 , , a n are given by …
    18: 3.8 Nonlinear Equations
    §3.8(iv) Zeros of Polynomials
    The polynomialFor further information on the computation of zeros of polynomials see McNamee (2007). … For describing the distribution of complex zeros of solutions of linear homogeneous second-order differential equations by methods based on the Liouville–Green (WKB) approximation, see Segura (2013). …
    Example. Wilkinson’s Polynomial
    19: 34.3 Basic Properties: 3 j Symbol
    §34.3(vii) Relations to Legendre Polynomials and Spherical Harmonics
    For the polynomials P l see §18.3, and for the function Y l , m see §14.30. …
    34.3.20 Y l 1 , m 1 ( θ , ϕ ) Y l 2 , m 2 ( θ , ϕ ) = l , m ( ( 2 l 1 + 1 ) ( 2 l 2 + 1 ) ( 2 l + 1 ) 4 π ) 1 2 ( l 1 l 2 l m 1 m 2 m ) Y l , m ( θ , ϕ ) ¯ ( l 1 l 2 l 0 0 0 ) ,
    34.3.22 0 2 π 0 π Y l 1 , m 1 ( θ , ϕ ) Y l 2 , m 2 ( θ , ϕ ) Y l 3 , m 3 ( θ , ϕ ) sin θ d θ d ϕ = ( ( 2 l 1 + 1 ) ( 2 l 2 + 1 ) ( 2 l 3 + 1 ) 4 π ) 1 2 ( l 1 l 2 l 3 0 0 0 ) ( l 1 l 2 l 3 m 1 m 2 m 3 ) .
    20: Bille C. Carlson
    In his paper Lauricella’s hypergeometric function F D (1963), he defined the R -function, a multivariate hypergeometric function that is homogeneous in its variables, each variable being paired with a parameter. …Also, the homogeneity of the R -function has led to a new type of mean value for several variables, accompanied by various inequalities. …